It can be observed that the right hand side commute with
\(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }\)  by the same argument as in Theorem REF , so we have
\(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }^1 \subset (\widetilde{\mathcal {M}}_{\mathrm {c},\kappa })^{\prime }\) .
Since we already know that the modular operator for \((\widetilde{\mathcal {M}}_{\mathrm {c},\kappa })^{\prime }\) 
with respect to \(\Omega _{\mathrm {c}}\otimes \Omega _{\mathrm {c}}\)  is \(\Delta _{\mathrm {c}}^{-1}\otimes \Delta _{\mathrm {c}}^{-1}\)  from Proposition REF ,
it is immediate to see that \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }^1\)  is globally invariant under the modular group of
\((\widetilde{\mathcal {M}}_{\mathrm {c},\kappa })^{\prime }\)  with respect to \(\Omega _{\mathrm {c}}\otimes \Omega _{\mathrm {c}}\) . Then the two algebras coincide
since \(\Omega _{\mathrm {c}}\otimes \Omega _{\mathrm {c}}\)  is cyclic for \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }^1\)  by a standard application of Takesaki's theorem
[1]} (see [2]}).

Variations of Propositions REF  and REF  hold for an action of \({\mathbb {Z}}_N\)  with
trivial changes.
